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Full text of "Treatise On Analysis Vol-Ii"

66       XII   TOPOLOGY AND TOPOLOGICAL ALGEBRA

and u e x9 v e y, then p(u + v) :g p(u) + p(v) and therefore (as u + v e x + y)
p(x -f y) ^p(u) +p(v). Since (2.3.11)

inf   (p(u) + p(v)) = p(x) + p(y),

uex,vey

the proof of (i) is complete.

(ii) Let r be a real number > 0 and let U be the set of all x e E such that
p(x) < r. Then it follows immediately from the definition (12.14.8.1) that n(U)
is the set of all x e E/F such that p(x) < r. Since F is directed, the sets U, as/>
runs through F and r runs through the set of real numbers >0, form a
fundamental system of neighborhoods of 0 in E. Hence (ii) follows from the
definition of neighborhoods in E/F (12.11).

Whenever we consider E/F as a locally convex space, it is always the
quotient topology that is meant, unless the contrary is expressly stated.

(12.1-4.9)   Every closed vector sub space of a Frechet space is a Frechet space.
Every quotient of a Frechet space by a closed vector subspace is a Frechet space.

The first assertion follows from above and from (3.14.5), the second from
(12.14.8) and (12.11.3).

Remark

(12.14.10) Let E be a normed space and F a closed vector subspace of E.
Then

(12.14.10.1)                     ||x|i=inf||z||=</(0,x)

Z 6 X

(in the last expression, x is considered as a linear variety parallel to F in the
vector space E) is a norm on E/F which defines the quotient topology, by
virtue of (12.14.4), (12.14.8), and (12.11.2). Whenever we consider E/F as a
normed space, it is always this norm that is meant, unless the contrary is
expressly stated.

More particularly, suppose that E is a Hilbert space. Then if F is a closed
subspace of E, and if F' = Pp 1(G) is the orthogonal supplement of F (6.3.1),
the bijective linear mapping F' -> E/F, the restriction to F' of the canonical
mapping n: E->E/F, is an isometry. For by Pythagoras' theorem we have
\\x' + z\\2 = \\x'\\2+ \\z\\2 for all zeF and all jc'eF; hence \\tf\\ is the
infimum of the norms of elements of x' -f- F. We may therefore identify E/F
with the Hilbert space F'.t it remains to prove the triangle inequality. If x, y are elements of E/F there exists a dense